I nternat. J. Math. & Math. Sci.
Vol. 5 No. 3 (1982) 613-619
613
SOLUTIONS OF TYPE r m FORA CLASS OF
SINGULAR EQUATIONS
ABDULLAH ALTIN
Faculty of Science
University of Ankara
Besevler, Ankara, TURKEY
(Received July 28, 1981)
ABSTRACT.
We obtain all solutions of radial type for a class of singular partial
differential equations of even order.
The essential operators here are elliptic or
ultrahyperbolic.
KEY WORDS AND PHRASES. Iterated elliptic or ultrahyperbolic equons, solutions
radial type, Lo.rentzian distance, hypernoidl domain, homogeneo polynomial
sol,ion, osory solution.
f
1980 MATHEMATICS SUBJECT CLASSIFICATION CODES.
i
35A08, 35J99, 35L99.
INTRODUCT ION
This paper concerns solutions of type r
m
for the class of partial differential
equat ions
P
]I
qj
L.
3
j=l
where p and
ql"’’’qp
qj
The iterated operators L.
(i.i)
0
+
+
Yi
Yi
+i2
r
(i.2)
are defined by the relations
Lo+l u
J
i= 1
u
P
are positive integers and
a
i
_+
xi xl xi
Lj
Lqp
ql
L1
u
n,
r
2
L
u
L (u)],
J J
-
(J)
i
i=l
I
i
x
s
2 +
i
i
U
2
i--i
Yi
,qj-l.
and T are any real parameters and
j
x
2 + 2
y
(i.3)
A. ALTIN
614
where x
Xn)
(x I
s
n
denote points in R and R
ys
(YI’
and y
respectively
are elliptic or ultrahyperbolic with the sign positive or negative,
The operators L
Equation (i.I) includes iterated forms of some well known classical
respectively.
equations such as the Laplace equation, the wave equation, and the EPD and GASPT
equations.
Many of these equations were studied by many authors in solving some
physical problems or extending known results [1-6].
m
Before we find solutions of type r of equation (I.i), we note that if the
operators
L. are ultrahyperbolic then, since the Lorentzian distance
x
r
Ixl
is not real for
IYl,
<
2
x
(iy)2
+
2
y
m
solutions of type r
i
are valid only in the hyperconoidal
domain
x D
D
n
where
Ixl
2
n
and R
domains centered at the origin in R
IYl
Ds
y
+ Ys2
s,
n
since r
D
x
x2n IYl 2
Yl +
+
I +
x
{(x,y)
s
<
Ixl
and D and D define spherical
s
n
respectively.
We also note that,
0 on the hypercone
(Dn x
solut ions of
type r
m
{(x,y)
Ds)
D s, x
Dn, y
x
2
y
2}
have singularities at the points of this hypercone surface.
In the case where the operators L. are elliptic, the regularity domain of
3
solutions of r
m
n+s
is a spherical domain centered at the origin in R
However, we shall see that equation (i.i)
O.
there is only a singularity at r
In this case,
has some polynomial solutions which are regular at all points.
2.
SOLUTIONS OF TYPE r
m
We first establish the following lemma.
LEMMA i.
Let
p
H
j=l
ql
qj
Lj
,qp be
(rm)
H
H
j=l k=O
{(m-
(m-2[Q(p)
where Q(j)
ql +
Then
any positive integers.
p qj-i
+ qj’
i
-< j
2.J
n
+
s
2[Q(p)
2k
Q(j)]
Q(j)]
+
2k)
2j)+ yj} rm-2Q(p)
(2 .i)
< p, and
2
+
e(J)
J
+
8
i--1
j)
(2.2)
615
SOLUTIONS OF A CLASS OF SINGULAR EQUATIONS
From the definition of L and r, we have
PROOF.
L. (r m)
Applying the operator L
[m(m + 2
+ y_]
r
m-2
(2.3)
repeatedly on both sides of (2.3), we then obtain by
induction the result
L
q. (rm)
3
q-i
N
[(m
2k)(m
and q in (2.4) by p and
Now replace
2k
+
k=O
qp,
+
2j)
yj]
r
m-2q
(2.4)
respectively, and apply in succession
the operators
Lqp-1
Lqp-2
p-2
p-i
on both sides of (2.4).
L
ql
1
By induction, we readily obtain formula (2.1).
Using Lemma i, we can now prove
I0,
parts
I
and
1
12,
I,
yj
0}
l
lj
’ j -Yj
> 0}
12
{j
0
r
m
jl
+
I
2
(I)
kj
PROOF.
be separated into three
e
2
I, j
< 0}
-yj
of equation (i.i) are given by the formula
r
k=0
[C(1)kj + C(2)kj
log r]
k=O
2 Q(p)-Q (j)
qj-i
Z
jel
where C
0
j
2
qj-I 2[Q(p)-Q(j)]-j+2k
Z
2
{j
Then, solution of the type u
jl
,p}
1,2
such that
I
u(r)
{j
Let the index set I
THEOREM i.
Z
k=0
C
%kj
(2)
%kj
kj
]-j+2k
r
cos
and
kj
2
j
log r +
j
kj
(2.5)
are arbitrary constants.
In Lemma i, if we set
m
2[Q(p)
then (2.1) can be written as
P
qJ )(rm)
(j=IH L.3
P
Q(j)]
2k
(2.6)
M,
qj-I
j=l k=0H
[M(M +
2j)
+
yj]
r
m-2Q(p)
"27)
A. ALTIN
616
Since the roots of the algebraic equation
M(M +
are M
@j
_+
j
+
2j)
(2.8)
0
j
we have from (2.6) the roots for m as
yj,
(1)
mkj
j
2[Q(p)
Q(j)]
Sj
+ 2k +
$
2[Q(p)
Q(j)]
Sj
+ 2k-
$
yj
(2)
rm-2Q(p)
yj
Thus, (2.7) can be written as
P
j=l
qJ
ej
qj-I
P
(r m)
N
N
j=l k=O
(i)
(m
from which it is easily seen that, for j
r
and r
mkj
q -i
p
j_
mkj
(2.9)
0,...,q.-l, the functions
l,...,p and k
Since the given equation is linear,
satisfy the given equation.
the sum
(m-
mkj
(2)
(i)
[Ukj rmkj
-(i)
mkj
C(2)
r
kj
+
(2. i0)
also satisfies (i.i).
If j
(
Ii,
then (2.8) has two distinct real roots.
Thus, the corresponding
solution of (2.10) will be
qj-I 2[Q(p)-Q(j)]-j
jeI 1 k=O
If j
12,
then (2.8) has complex roots.
r+i
J
J
e+i--yJ
cos
(yj
Sj
2
J
(1)
ckj
r
+
r
(2)
Ckj
J
J
(2. ii)
r
By the properties
log r
$j
2
log r) + i sin
$j
j
log r),
the corresponding solution of (2.10) will be
Z12
k=O
j(
where C
(I)
kj
If j
+
C
(2)
kj
Io,
2 [Q(p)-Q(j)
qj-i
%kj
%kj
]-@j+2k
r
co s
cos
kj
and
(1)
c kj
then (2.8) has double root
(2)
ckj
(0)
mkj
yj-$j
2
i
j
log r
sin
that is,
kj"
+ kj
(2.12)
617
SOLUTIONS OF A CLASS OF SINGULAR EQUATIONS
mkj(1) mkj(2)
2[Q(p)
+
j
Q(j)]
So the right hand side of (2.9) has the factors [m-
(0)
r
which itself and its
Thus, each of the func-
_(0)
mkj
d
m
and
(rm)
2 [Q(p)-Q(j)
qj-i
rmkj
Ira--(0)
-"j
log r
Hence, the corresponding solution of (2.10) will be
satisfies the given equation.
J10 kO
O) 2
j
j0.).
first derivative with respect to m are zero for m
tions
j0)
2k
]-j+2k
C(1)
kj
r
(2)
+ Ckj
log r]
Therefore, the sum of (2.11), (2.12), and (2.13) gives (2.5).
(2.13)
Thus, the theorem
is proved.
REMARK i.
the root M
In the special case where
0, the algebraic equation (2.8) has
yj
In this case, since the values
0.
2[Q(p)
mkj
are nonnegative integers for j
Q(j)]
l,...,p and k
r
+ 2k
0,...,qj-l,
the functions
2[Q(p) -Q(j)] + 2k
are homogeneous polynomial solutions of equation (i.I).
From (2.5), we see that
it is not possible to obtain polynomial solutions for equation (i.I) in all cases
where
# 0.
yj
It is possible, however, if the following condition is satisfied.
If the algebraic equation (2.8) has integral roots M
u
for some j
e
I,
such that
nD
0,...,qD-l,
for k
of equation (i.I).
M
u
>
2[Q(p)
Q(D)] + 2k +
MD
>0
then the functions r mkD are homogeneous polynomials solutions
It is clear that the above inequality is always satisfied if
0.
REMARK 2.
From (2.5), we see that, if j
e
12,
that is, the algebraic equation
(2.8) has complex roots, equation (i.i) then has oscillatory solutions in the re.gularity domain of its solutions.
THEOREM 2.
by formula (2.5)
All solutions of type u
f(r) for equation (i.i) can be expressed
618
A. ALTIN
PROOF.
Consider operator (1.2).
By direct calculation, it is easily verified
that
j
dr
which is an Euler type operator.
d
e
d-
-t
2
If we set r
D
d
and
e
2
(2.14)
and D
e
2
dr
+ 2
r
dr
r
-2t
(D
then we have
2
D)
Hen ce,
L.j(u)
If we let
Fj(D)
D
2
+
e
2#jD + i.’
-2t
(D
2
+
2jD + yj)
(2.15)
u
then (2.15) may be written as
e.(u)
e
-2t
F.(D)
(2.16)
u
From ordinary differential equations, we know that, for any polynomials with constant coefficients G and H and for any constant
-at
G(D) {e
H(D) u}
e
-at
a, the following relation is valid
G(D- a) H(D) u.
Considering the properties of (2.17) and applying the operator
(2.17)
L. repeatedly
on
3
both sides of (2.16), we then obtain by induction the result
q.
L
3
(u)
e
-2qt q-i
N
k=O
F.(D
2k) u.
3
We remark that the product of the operators
place j and q in
(2.18i
by p and
qp,
F. (D)
(2.18)
are commutative.
Now re-
respectively, and apply in succession the op-
erators
Lqpp-i i Lqpp-2 2
on both sides of (2.18).
P
qj
j=l Lj
u
q
LI
i
By induction, we obtain the formula
e
-2Q(p)t
P
qj-i
j=in k=ON Fj(D
2k
2[Q(p)-Q(j)]) u
(2.19)
Equating this last expression to zero, we obtain an ordinary differential equation
with constant coefficients and of order 2Q(p)
2(ql
+
+
equation for this equation is
P
qj-I
j=l k=O
F (m- 2k- 2[Q(p)-Q(j)])
J
0
qp).
The indical
619
SOLUTIONS OF A CLASS OF SINGULAR EQUATIONS
M in this equation, we find
Substituting m- 2k- 2[Q(p)-Q(j)]
P
qj-i
H
H
[M(M +
j=l k=0
+
2j
yj]
(2.20)
0.
This was obtained previously on the right hand side of (2.7).
It is obvious that
the corresponding solution for this equation is given by (2.5).
We note that, if we substitute u-- r
and e
-2Q(p)t
r
-2 Q (p)
m
in (2.19) then, by considering r
P
qj-i
H
H
F.(D
e
2[Q(p)-Q(j)]) e
2k
mt
P
mt
qj-i
j=l k=0
se
mt
e
and
j--1 k--0--
we
m
F (m
J
2k
2[Q(p)-Q(j)])
that (2.19) reduces to (2.1).
REFERENCES
i.
ALMANSI, E.
2.
ALTIN, A. and YOUNG, E.C.
3o
ALTIN, A.
4.
DIAZ, J.B. and YOUNG, E.C.
5.
MILES, E.P. and YOUNG, E.C.
6.
WEINSTEIN, A.
Annali di Mathematica, serie
II,III (1890), pp. 1-59.
Some Properties of Solutions of a Class of Partial
Differential Equations (to appear).
Some Expansion Formulas for a Class of Singular Partial Differential
Equations, Proc. Amer. Math. Sock., 1982.
A Singular Characteristic Boundary Value Problem
for the Euler-Poisson-Darboux Equation, Ann. di Mat. Pura ed Appl.,
IV 95, pp. 115-129 (1972).
Basic Sets of Polynomials for Generalized Beltrami
and Euler-Poisson-Darboux Equations and Their Iterates, Proc. Amer. Math.
So___c. 18, (1967), pp. 981-986.
On a Class of Partial Differential Equations of Even Order,
Annali di Mat. 3_9, pp. 245-254 (1955).